1621. Number of Sets of K Non-Overlapping Line Segments

https://leetcode.com/problems/number-of-sets-of-k-non-overlapping-line-segments

Description

Given n points on a 1-D plane, where the ith point (from 0 to n-1) is at x = i, find the number of ways we can draw exactly k non-overlapping line segments such that each segment covers two or more points. The endpoints of each segment must have integral coordinates. The k line segments do not have to cover all n points, and they are allowed to share endpoints.

Return the number of ways we can draw k non-overlapping line segments*.* Since this number can be huge, return it modulo 109 + 7.

Example 1:

**Input:** n = 4, k = 2
**Output:** 5
**Explanation:**The two line segments are shown in red and blue.
The image above shows the 5 different ways {(0,2),(2,3)}, {(0,1),(1,3)}, {(0,1),(2,3)}, {(1,2),(2,3)}, {(0,1),(1,2)}.

Example 2:

**Input:** n = 3, k = 1
**Output:** 3
**Explanation:** The 3 ways are {(0,1)}, {(0,2)}, {(1,2)}.

Example 3:

**Input:** n = 30, k = 7
**Output:** 796297179
**Explanation:** The total number of possible ways to draw 7 line segments is 3796297200. Taking this number modulo 109 + 7 gives us 796297179.

Example 4:

**Input:** n = 5, k = 3
**Output:** 7

Example 5:

**Input:** n = 3, k = 2
**Output:** 1

Constraints:

• 2 <= n <= 1000

• 1 <= k <= n-1

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